The Steinhaus-Weil property: I. Subcontinuity and amenability

Bingham, Nick H. and Ostaszewski, Adam ORCID: 0000-0003-2630-8663 (2020) The Steinhaus-Weil property: I. Subcontinuity and amenability. Sarajevo Journal of Mathematics, 16 (1). 13 - 32. ISSN 1840-0655

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Abstract

The Steinhaus-Weil theorem that concerns us here is the simple, or classical, 'interior-points' property - that in a Polish topological group a non-negligible set B has the identity as an interior point of Bˆ-1B: There are various converses; the one that mainly concerns us is due to Simmons and Mospan. Here the group is locally compact, so we have a Haar reference measure η. The Simmons-Mospan theorem states that a (regular Borel) measure has such a Steinhaus-Weil property if and only if it is absolutely continuous with respect to the Haar measure. This the first of four companion papers (we refer to the others as II [BinO11], III, [BinO12], and IV, [BinO13], below). Here (Propositions 1-7 and Theorems 1-4) we exploit the connection between the interior-points property and a selective form of infinitesimal invariance afforded by a certain family of selective reference measures σ, drawing on Soleckiís amenability at 1 (and using Fuller's notion of subcontinuity). In II, we turn to a converse of the Steinhaus-Weil theorem, the Simmons- Mospan theorem, and related results. In III, we discuss Weil topologies, linking the topological group-theoretic and measure-theoretic aspects. We close in IV with some other interior-point results related to the Steinhaus- Weil theorem.

Item Type:	Article
Official URL:	http://www.anubih.ba/Journals/SJMATH.html
Additional Information:	© 2020 Department of Natural Sciences and Mathematics, Academy of Sciences and Arts of Bosnia and Herzegovina (ANUBIH).
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	09 Mar 2020 13:48
Last Modified:	12 Dec 2024 02:05
URI:	http://eprints.lse.ac.uk/id/eprint/103708

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